Continuity of measurable cocycles

TL;DR

This paper proves that under certain measurability conditions, a cocycle for a Polish group action that is continuous in the second variable must be jointly continuous, extending the understanding of cocycle regularity.

Contribution

It establishes joint continuity of cocycles under Baire or measure measurability assumptions, generalizing previous results on cocycle regularity in Polish group actions.

Findings

Cocycles continuous in the second variable are jointly continuous under certain measurability conditions.

The result applies to Baire measurable and measure measurable cocycles.

Provides a new criterion for cocycle regularity in the context of Polish group actions.

Abstract

Suppose $G\curvearrowright X$ is a Polish group action, $H$ is a Polish group and $G\times X\overset{\psi}\longrightarrow H$ is a cocycle that is continuous in the second variable. If $\psi$ is either Baire measurable or is $\lambda\times \mu$-measurable with respect to a Haar measure $\lambda$ on $G$ and a fully supported $\sigma$-finite Borel measure $\mu$ on $X$, then $\psi$ is jointly continuous.